I have heard that Sinai's famous results on the ergodicity of $n$ hard spheres defined on a $d$ dimensional hypercube with periodic boundaries is isomorphic to a billiard (from which it was proved), now known as the Sinai billiard. That is, a billiard again defined on a $d$ dimensional flat torus except with a solid disk/cylinder of radius $r$ removed from its centre. The informal explanation for this is that the collision of hard spheres occurs when the centres of the spheres are within a distance $2r$ from each other. So, somehow the collisions of point/billiard particles with this circular scatterer is equivalent to a collision of two hard spheres?
What is the mathematical explanation of this isomorphism? In particular, how does one reconcile the fact that with $n$ hard spheres, there are more than just 'one' collision happening at once? i.e. $3$ sphere collisions or indeed multiple $2$ sphere collisions not happening about some central point (as the billiard construction seems to suggest). I suspect part of the isomorphism involves 'stripping away' some of these complicated 'physical' effects, but it is hard to imagine how. Is it also true that ergodic problem involving just two hard spheres is equivalent to that involving $n$ of them?